| dc.description.abstract | This thesis develops some general calculational techniques for finding the orders of knots
in the topological concordance group C . The techniques currently available in the literature
are either too theoretical, applying to only a small number of knots, or are designed to only
deal with a specific knot. The thesis builds on the results of Herald, Kirk and Livingston
[HKL10] and Tamulis [Tam02] to give a series of criteria, using twisted Alexander polynomials,
for determining whether a knot is of infinite order in C.
There are two immediate applications of these theorems. The first is to give the structure
of the subgroups of the concordance group C and the algebraic concordance group G generated
by the prime knots of 9 or fewer crossings. This should be of practical value to the knot-theoretic
community, but more importantly it provides interesting examples of phenomena
both in the algebraic and geometric concordance groups. The second application is to find
the concordance orders of all prime knots with up to 12 crossings. At the time of writing of
this thesis, there are 325 such knots listed as having unknown concordance order. The thesis
includes the computation of the orders of all except two of these.
In addition to using twisted Alexander polynomials to determine the concordance order
of a knot, a theorem of Cochran, Orr and Teichner [COT03] is applied to prove that the n-twisted doubles of the unknot are not slice for n ≠ 0,2. This technique involves analysing the
`second-order' invariants of a knot; that is, slice invariants (in this case, signatures) of a set of
metabolising curves on a Seifert surface for the knot. The thesis extends the result to provide
a set of criteria for the n-twisted double of a general knot K to be slice; that is, of order 0 in C.
The structure of the knot concordance group continues to remain a mystery, but the thesis
provides a new angle for attacking problems in this field and it provides new evidence for
long-standing conjectures. | en |
